Konstantin Recke (Oxford University) – Vanishing uniqueness thresholds for Poisson--Voronoi percolation

Poisson-Voronoi percolation can be defined on any metric space $(M,d)$ with an infinite Radon measure $\mu$ as follows. For $\lambda>0$, consider a Poisson point process of intensity $\lambda\mu$ and associate to each point of the process its Voronoi cell, that is the set of all points in $M$ closer to this point than to any other point of the process. For $p\in(0,1)$, independently color each cell black with probability $p$ and consider the union of black cells.

For fixed $\lambda$, there is a phase transition reminiscent of classical Bernoulli percolation with a critical probability $p_c(\lambda)$, which is the infimal value of $p$ such that an unbounded cluster exists, and a possible different uniqueness threshold $p_u(\lambda)$, which is the infimal value of $p$ such that a unique unbounded cluster exists.

In this talk, I will review results about this model in Euclidean space $\mathbb R^d$ and also hyperbolic space $\mathbb H^d$ where interesting behavior of the functions $p_c(\lambda)$ and $p_u(\lambda)$ emerges. I will then discuss a new phenomenon, namely the fact that $p_u(\lambda)$ can become arbitrarily small, and its applications, in several situations of interest.

The talk is based on joint works with Jan Grebík and with Matteo D'Achille, Jan Grebík, Ali Khezeli and Amanda Wilkens.